On Chevalley restriction theorem

Abstract

Let $g$ be a complex semisimple Lie algebra with adjoint group $G$. Suppose that $\sigma$ is an involutive automorphism of $g$. Then $\sigma$ induces uniquely an involution of $G$ also denoted by $\sigma$, let $K=G^\sigma$ be a subgroup of $\sigma$-fixed points. Consider a direct decomposition $g=k+p$ of $g$ into eigenspaces for $\sigma$. Then $p$ is a $K$-module. Denote by $a\subset p$ any maximal abelian ad-diagonalizable subalgebra. Consider the ``baby Weyl group'' $W=N_K(a)/Z_K(a)$. Let $\psi: C[p]^K\to C[a]^W$ be a restriction map of algebras of invariants. Then the famous Chevalley restriction theorem states that $\psi$ is an isomorphism. The aim of this paper is prove the following Theorem. The restriction map $\psi: C[p\times p]^K\to C[a\times a]^W$ is surjective.